7. Übungsblatt zur Vorlesung WS 2019/20
Einführung in die Elementarteilchentheorie Prof. G. Hiller Abgabe: bis Di. 26. November 2019, 12:00 Uhr Kasten 245 http://people.het.physik.tu-dortmund.de/~ghiller/WS1920ETT.html
Exercise 1: Bhabha scattering (12 Points)
Compute the differential cross sectiondσ/dcosθfor the processe+e−→e+e−using the QED Feynman rules at leading order. Here,θdenotes the scattering angle in the center of mass frame of the incoming fermions. The electron mass can be neglected in this calculation.
(a) Draw all lowest order QED Feynman diagrams for this process.
(b) Give the corresponding matrix elementsMi for each diagram and the total matrix elementM=P
iMi.
(c) Use your results from (b) to compute the spin-averaged squared matrix element
|M|2= 1
(2S1+1)(2S2+1) X
Spi ns
|M|2, (1.1)
whereS1andS2denote the spins of the initial state particles.
(d) Show that the differential cross section is given as dσ
dcosθ =πα2 s
·u2+t2 s2 +2u2
st +u2+s2 t2
¸
, (1.2)
wheres,t anduare the Mandelstam variables andα=e2/(4π).
(e) Why and how can you relate the result from (d) to the differential cross section for Møller scatteringe−e−→e−e−
dσ
dcosθ=πα2 s
·s2+t2 u2 +2s2
ut +u2+s2 t2
¸
? (1.3)
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Exercise 2:e+e−→π+π−pion production (8 Points)
Consider the following reaction:
e+(k+)e−(k−)→π+(p+)π−(p−) , (2.1) in the center-of-mass frame of the leptons. You can neglect the masses of the leptons.
The pionsπ±are hadrons with spin 0 and charge±1e . The mass of the pion ismπ± = 139.57 MeV. The matrix element of the hadronic current reads
Jµπ+π−=a(q2)(p+−p−)µ+b(q2)(p++p−)µ, (2.2) wherea(q2)andb(q2)are formfactors andq2=(p++p−)2.
(a) Draw the lowest order QED Feynman diagram for the process in Eq. (2.1). Why is Eq. (2.2) already the complete expression for the hadronic current, that is, why are there no other terms in this equation?
(b) Imposing current conservationqµJµπ+π−=0, what does this mean for the form fac- tors?
(c) The Feynman rule for the photon coupling to pions reads+ie Jµπ+π−. The rule for external scalars is simply 1. Determine the matrix element for the process in Eq.
(2.1) up to leading order.
(d) Calculate the differential cross sectiondσ/d cosθ, whereθdenotes the scattering angle betweene−andπ−. Calculate the total cross section in the high energy limit.
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